How can I make this portfolio self-financing?

Question

$a_t S_t$ = number of shares ($S_t$ is stock price at $t$), $S_0 = 1$

$b_t \beta _t$ = saving account value , $d \beta_t = r \beta_t dt$, $r=$ interest rate

So the value of the portfolio:

$$V_t = a_t S_t + b_t \beta_t$$

Is self-financing if

$$dV_t = a_t dS_t + b_t d \beta_t$$

If $a_t = 1-t$, how can I choose $b_t$ such that my portfolio is self-financing?

$$V_t = (1-t)S_t + b_t \beta_t$$

How do I formulate $dV_t$ now? Don't I require more information, in particular, what is $S_t$?

Is there a need to use the stochastic product rule?

Answer 1

We have $$ V_t = a_t S_t + b_t \beta_t. $$

By Ito's product rule, \begin{align*} dV_t & = d(a_t S_t) + d(b_t \beta_t) \\ & = a_t dS_t + S_t da_t + da_t dS_t + b_t d\beta_t + \beta_t db_t + db_td\beta_t. \end{align*}

Since $da_t$ and $db_t$ have no $dW_t$ term, the cross terms are both zero and we have \begin{align*} dV_t & = a_t dS_t + S_t da_t + b_t d\beta_t + \beta_t db_t. \end{align*}

Now just plug in your value for $da_t$ and solve one equation in the unknown $b_t$: \begin{align*} dV_t & = a_tdS_t - S_t dt + b_t d\beta_t + \beta_t db_t \triangleq a_t dS_t + b_t d\beta_t \\ & \iff \beta_t db_t = S_t dt \\ & \iff b_t = b_0 + \int_0^t S_u/\beta_u du. \end{align*}

Now you may be able to solve for $b_t$ explicitly depending on your model for $S_t$.

Answer 2

In the Black-Scholes model, you would have $d S_t = \mu\, d t + \sigma\, d W_t$ where $W$ is a Brownian motion. So if $V_t = a_t S_t + b_t \beta_t$, then $$ dV_t = a_t\, d S_t + S_t\, d a_t + da_t\,dS_t + b_t\,d\beta_t + \beta_t\,d b_t + db_t\, d\beta_t $$ by the product rule. In your case, when $a_t = 1-t$ you will have $$ dV_t = (1-t) \, dS_t - S_t\, dt + b_t\,d\beta_t + \beta_t\,db_t $$ since $da$ and $d\beta$ have no $dW$-term. Hence, you will need to pick $b$ such that $\beta\,db_t = S_t\,dt$, for the portfolio to be self-financing.